IntroductionIn mathematics and theoretical computer science, a fixed point of an operator $F$ is an entity $x$ such that $F(x) = x$. Fixed-point results appear across many domains: for example, Banach’s Fixed-Point Theorem guarantees a unique fixed point for any contraction mapping on a complete metric space, and the Knaster–Tarski Theorem ensures that every monotone function on a complete lattice has a fixed point. These classical theorems establish existence (and sometimes uniqueness) of solutions to $x = F(x)$ under various conditions. In recursive and computational settings, fixed points enable self-referential definitions – a recursive function can be seen as a fixed point of a functional that “unwinds” one step of the recursion. For instance, the Y-combinator in lambda calculus provides a fixed-point combinator $Y$ such that for any function $W$, $Y(W)$ is a term satisfying $Y(W) = W(Y(W))$. This yields recursive definitions (like the factorial function) as solutions to self-referential equations.This article develops a fully formal framework for recursive fixed points – fixed points obtained via an iterative or recursive process. We focus on the convergence of a sequence (possibly transfinite) of transformations to a self-consistent state. Intuitively, we start with an initial approximation and repeatedly apply a transformation $\phi$; if this process approaches a stable state that no longer changes under $\phi$, we have reached a fixed point. Formally, one may consider an iterative sequence $x_0, x_1 = \phi(x_0), x_2 = \phi(x_1), \dots$ and seek a limit $x_{\infty}$ such that $\phi(x_{\infty}) = x_{\infty}$. Such a limit, if it exists, is a recursive fixed point – the result of infinitely (or transfinitely) many applications of $\phi$. This idea can be generalized beyond simple sequences, using the machinery of ordinal-indexed recursion and category theory to rigorously construct $\phi^\infty$, the outcome of transfinitely many iterations of $\phi$.Recent research by Alpay (2025) introduced Alpay Algebra, a category-theoretic framework where transfinite recursive fixed points play a central role. In this framework, an endofunctor $\phi$ (an operator on objects in a category) can be iterated through ordinal numbers to yield a stable initial fixed point denoted $\phi^{\infty}$. The existence of $\phi^{\infty}$ under broad conditions and its universal properties have been proven with mathematical rigor. The fixed point $\phi^{\infty}$ is recursive in that it is obtained as the limit of an ordinal-indexed chain of iterative approximants (often called the initial chain). Notably, this $\phi^{\infty}$ is not just any solution to $X \cong \phi(X)$; it is the smallest or initial solution, meaning it is generated by the recursive process itself and any other fixed point admits a unique morphism from $\phi^{\infty}$. In other words, $\phi^{\infty}$ encapsulates the “ultimate outcome” of the transformation $\phi$ applied repeatedly without end – a self-consistent structure that remains invariant under $\phi$.This manuscript provides a formal exposition of recursive fixed points. We begin by establishing the mathematical preliminaries (category-theoretic foundations and definitions of transfinite iteration). We then prove the existence and uniqueness of the transfinite fixed point $\phi^{\infty}$ under appropriate conditions, drawing on recent developments in Alpay Algebra. We illustrate these concepts with examples ranging from classical structures (natural numbers, infinite streams) to logic (fixed-point semantics of recursive theories) and AI systems (iterative embedding alignment) to demonstrate the ubiquity of recursive fixed points in theory and practice. Throughout, we use a formal style with symbolic notation – emphasizing symbols over prose – to maximize precision and semantic weight. By the end, we will see that recursive fixed points not only exist, but in fact serve as universal invariants in many self-referential systems, providing a rigorous backbone for understanding phenomena like emergent consistency, identity of processes, and convergence of iterative algorithms.Preliminaries: Category-Theoretic FrameworkOur development uses the language of category theory to formalize recursive processes. We briefly summarize the needed notions (for a comprehensive background, see Mac Lane, 1971). We assume a category $\mathcal{A}$ with an initial object $\varnothing$ (an object with a unique morphism from it to any other object). An endofunctor $\phi: \mathcal{A} \to \mathcal{A}$ is an operator that maps objects to objects and morphisms to morphisms within $\mathcal{A}$. Intuitively, $\phi$ represents one step of a generative or transformative process on the structures in $\mathcal{A}$. We are interested in objects $X$ that satisfy an isomorphism $X \cong \phi(X)$ – such objects are fixed points of the functor $\phi$ (also called $\phi$-algebras that are self-consistent).Transfinite Ordinals and Chains: To capture recursive (potentially infinite) iteration, we consider ordinal numbers $0, 1, 2, \dots, \omega, \omega+1, \dots$ which extend the natural numbers into the transfinite. An ordinal-indexed chain(or transfinite sequence) in $\mathcal{A}$ is a family of objects ${X_{\alpha}}{\alpha < \lambda}$ for some ordinal $\lambda$, together with morphisms connecting them, such that $X{0} = \varnothing$ (the initial object), and for each ordinal $\beta < \lambda$:Successor step: If $\beta = \alpha+1$ is a successor, then $X_{\beta} = \phi(X_{\alpha})$. In other words, each step applies the functor $\phi$ to the previous object.Limit step: If $\beta$ is a limit ordinal (zero is the minimal ordinal, any non-zero ordinal with no immediate predecessor is a limit), then $X_{\beta}$ is defined as the colimit (categorical limit of the diagram) of all earlier $X_{\alpha}$ for $\alpha < \beta$. Intuitively, at a limit stage, $X_{\beta}$ is the “union” or limit of the prior approximations $X_{0}, X_{1}, ..., X_{\alpha}, (\alpha<\beta)$.This process yields an initial chain:X0→ X1=ϕ(X0)→ X2=ϕ2(X0)→ ⋯→ Xω=colim{Xn:n<ω}→ Xω+1=ϕ(Xω)→ ⋯Each stage $X_{\alpha}$ is built “recursively” from the previous ones. We say $\phi$ is continuous (or $\kappa$-accessible) if it preserves colimits of chains of length $<\kappa$ for some regular cardinal $\kappa$ (for example, $\omega$-continuous means it preserves countable colimits). Under such conditions, one can show that the initial chain eventually reaches a stage where applying $\phi$ does not produce a new object. Formally, there exists some ordinal $\mu$ (often $\mu = \kappa$ or earlier) such that $X_{\mu} \cong X_{\mu+1} = \phi(X_{\mu})$. When this first occurs, $X_{\mu}$ is a fixed point of $\phi$. By construction, it is the minimal or initial fixed point, since it arose from the smallest starting object by iterative application of $\phi$. We denote this object as $\mu \phi$ or $\phi^{\infty}$ (Alpay’s notation). It is also called the initial $\phi$-algebra in category-theoretic terms.Definition: The recursive fixed point of $\phi$, denoted $\phi^{\infty}$, is the object (if it exists) at which the transfinite iterative chain stabilizes. Concretely, $\phi^{\infty}$ is an object such that $\phi(\phi^{\infty}) \cong \phi^{\infty}$ and for some ordinal $\mu$, $\phi^{\infty} = X_{\mu}$ with $X_{\mu} \cong \phi(X_{\mu})$, where ${X_{\alpha}}$ is the initial chain defined above.Because $\phi^{\infty}$ arises by iterating $\phi$ starting from the smallest object, it intuitively represents the “limit of applying $\phi$ forever.” This aligns with the idea of a recursive process converging to a fixed point. The existence of $\phi^{\infty}$ is not automatic in every category or for every functor $\phi$; it typically requires conditions like completeness of the category or continuity of $\phi$ as mentioned. The Alpay Algebra framework explicitly assumes such conditions (e.g., working in well-behaved categories with transfinite colimits and $\phi$ preserving those colimits). In fact, Alpay Algebra I establishes that $\phi^{\infty}$ exists for every initial object under ZFC set theory assumptions without additional axioms. All proofs are carried out within standard category-theoretic foundations (following Mac Lane’s paradigm).Existence of the Recursive Fixed Point ($\phi^{\infty}$)We now state and prove (in outline) the Existence Theorem for recursive fixed points. This corresponds to the fixed-point existence results found in Alpay’s work and is analogous to the classical results (Knaster-Tarski, etc.) but in a transfinite categorical setting.Theorem 1 (Existence of $\phi^{\infty}$): Let $\phi: \mathcal{A} \to \mathcal{A}$ be an endofunctor on a category $\mathcal{A}$ that admits all colimits of chains of length up to some regular ordinal $\Lambda$, and assume $\phi$ preserves these colimits (i.e. $\phi$ is $\Lambda$-continuous). If $\mathcal{A}$ has an initial object $X_0$, then the initial chain $(X_{\alpha}){\alpha < \Lambda}$ defined by $X{0} = \varnothing$ and $X_{\alpha+1} = \phi(X_{\alpha})$ (with $X_{\lambda} = \mathrm{colim}{\alpha<\lambda} X{\alpha}$ for limit ordinals $\lambda < \Lambda$) will converge to a fixed point. In other words, there exists some ordinal $\mu < \Lambda$ such that $X_{\mu} \cong \phi(X_{\mu})$. This object $X_{\mu}$ is the recursive fixed point $\phi^{\infty}$. Moreover, $\phi^{\infty}$ is an initial algebra for $\phi$: the unique $\phi$-algebra generated by this transfinite iteration.Proof Sketch: Because $\Lambda$ is regular and $\phi$ preserves colimits of chains of length $<\Lambda$, the chain cannot continue to produce strictly larger (non-isomorphic) objects at every stage without end; if it did, one could take the colimit at stage $\Lambda$ (contradicting regularity or continuity). Thus there must be some stage where stabilization occurs. Formally, consider the sequence of inclusions (morphisms) $X_0 \to X_1 \to X_2 \to \cdots$. Either this sequence continues strictly (no stabilization) for all ordinals $<\Lambda$, or else there is a stage $\mu$ where $X_{\mu} \to X_{\mu+1}$ is an isomorphism. The former scenario is ruled out by a cardinality/cumulativity argument: if no stabilization occurs before $\Lambda$, then $X_{\Lambda} = \mathrm{colim}{\alpha<\Lambda} X{\alpha}$ is a fixed point of $\phi$ at stage $\Lambda$, because $\phi(X_{\Lambda}) = \phi(\mathrm{colim}{\alpha<\Lambda} X{\alpha}) \cong \mathrm{colim}{\alpha<\Lambda} \phi(X{\alpha}) = \mathrm{colim}{\alpha<\Lambda} X{\alpha+1} = \mathrm{colim}{\alpha<\Lambda} X{\alpha} = X_{\Lambda}$ (using continuity of $\phi$). Thus $X_{\Lambda}$ itself would be a fixed point, effectively $X_{\Lambda} \cong \phi(X_{\Lambda})$, achieving stabilization at $\Lambda`. In either case, we obtain some least ordinal $\mu$ (possibly $\mu=\Lambda$ if no earlier stage) such that $X_{\mu} \cong \phi(X_{\mu})$. Define $\phi^{\infty} := X_{\mu}$. By construction, $\phi^{\infty}$ satisfies $\phi(\phi^{\infty}) \cong \phi^{\infty}$. Furthermore, for any $\phi$-algebra $(X,\alpha: \phi(X)\to X)$ (any other fixed structure), we have by initiality of the chain that there is a unique homomorphism from each $X_{\alpha}$ into $X$ commuting with the $\phi$-action; at the limit, this yields a homomorphism $h: \phi^{\infty} \to X$. Thus $\phi^{\infty}$ is the initial object among all solutions of $X \cong \phi(X)$. ∎This theorem formalizes the existence of a transfinitely attained fixed point. In plainer terms, if one keeps applying $\phi$ starting from the simplest object, eventually (perhaps after an infinite number of steps) one stops getting new structures and hits a self-consistent one. That endpoint is $\phi^{\infty}$. All the intermediate steps $X_0 \to X_1 \to \cdots \to X_{\mu}=\phi^{\infty}$ are essentially building up a solution to the equation $X = \phi(X)$ piece by piece (like successive approximations). The condition of $\phi$ preserving colimits ensures that no information is lost in the limit process and that $\phi^{\infty}$ truly is a fixed point.In the context of Alpay Algebra (a formal system introduced by Faruk Alpay), Theorem 1 is a central pillar: it guarantees that for the self-referential processes defined in that framework, a stable identity emerges as a fixed point. Specifically, Alpay Algebra treats $\phi^\infty$ as representing the intrinsic identity of a generative process, since it is the unique invariant state that the process converges to. The existence theorem above matches statements in Alpay’s work such as: “We prove that the fixed point $\phi^\infty$ exists for every initial object ... and [that] $\phi$-iterates converge under regular cardinals”. The iterative construction of $\phi^\infty$ is sometimes called an ordinal-indexed fold or transfinite induction on the functor.It is worth noting that this categorical result generalizes classical fixed-point constructions. For example, in domain theory (a branch of theoretical computer science and math), a special case of this theorem states that any continuous endofunction on a complete partial order has a least fixed point (by taking the $\omega$-chain of iterates starting from the bottom element). That is essentially the $\omega$-continuous case of the above, corresponding to Kleene’s Fixed-Point Theorem for monotone operators on CPOs (which is itself an instance of Tarski’s theorem for lattices). Our transfinite approach extends this idea: even if $\omega$ steps are not enough, one allows transfinite steps until convergence. The result is a powerful guarantee: if the process can keep going without inconsistency, it will find a fixed point. In practical terms, this means any self-referential or recursive definition, under broad conditions, “bottoms out” at a well-defined semantics or structure that does not further change under the defining transformation.Uniqueness and Universal PropertyHaving established existence, we examine the uniqueness and universal property of the recursive fixed point $\phi^{\infty}$. Uniqueness here means $\phi^{\infty}$ is essentially the only fixed point that can be obtained through the recursive construction – if there were another built in the it would be to $\phi^{\infty}$. $\phi^{\infty}$ is the smallest fixed point, and every other fixed point of $\phi$ out” of $\phi^{\infty}$ in a unique and In the of Theorem 1, the recursive fixed point $\phi^{\infty}$ is unique up to isomorphism. Moreover, $\phi^{\infty}$ has the initiality for any object $X$ with an isomorphism \cong \phi(X)$ (i.e. any other fixed point of there exists a unique morphism \phi^{\infty} \to (a the appropriate In any other fixed-point object $X$ admits a unique from $\phi^{\infty}$, which is an isomorphism if $X$ is also obtained via a iterative Sketch: Uniqueness up to isomorphism is a standard category-theoretic argument: if and are initial $\phi$-algebras fixed then by initiality of there is a unique homomorphism I \to and by initiality of a unique homomorphism \to The I \to must the identity on (by uniqueness of the on and = Thus and are \cong This any solutions of the transfinite recursion are we can of the $\phi^{\infty}$ up to the universal any other fixed point we can $X$ with a $\phi$-algebra structure and to Specifically, \to can serve as the structure on \cong is there is a unique $\phi$-algebra homomorphism \phi^{\infty} \to such that = Intuitively, maps the fixed into any fixed structure $X$ in the only with $\phi$. If $X$ itself built by a iterative limit process (e.g., another chain that also a fixed then will be an isomorphism $\phi^{\infty}$, cannot into a end up the and formal terms, there is only one smallest self-consistent structure generated by $\phi$, and any other self-consistent structure a from this smallest one. This $\phi^{\infty}$ the universal fixed point or the identity of the process $\phi$. other solution of $X \cong \phi(X)$ through $\phi^{\infty}$ in a unique The morphism \phi^{\infty} \to can be seen as the or the fixed point $\phi^{\infty}$ into the fixed point If $X$ is not not be or but it exists and is universal property that $\phi^{\infty}$ the of all of the recursive process. In Alpay’s terms, $\phi^{\infty}$ as the of the a fixed point that every other fixed point in a This is analogous to the least fixed point in a lattice is in (or maps every other fixed point in that of Recursive Fixed illustrate the theory with recursive fixed points in various the functor = 1 on the category is a $\phi$-algebra is \to which provides a point (the of the and a function \to (the on the $X$ $X \cong 1 in yields the result that $X$ must be a infinite set to the natural numbers (with the point corresponding to and the function corresponding to the starting from the initial object and iterating $\phi$: $X_0 = = 1 \cong (a = 1 1 \cong = 1 \cong ... one \cong and the colimit as \to is \cong $\omega$ we have \cong 1 has a point and the is to is a fixed point of $\phi$. This is $\phi^{\infty}$ in this case – the recursive fixed point representing the defined natural number It is the smallest solution of $X = and any other of has a unique embedding from Let = for some fixed set $\phi$-algebra is \to which is like an on $X \cong in yields no solution the (if is there is no $X$ satisfying = if we infinite the equation $X \cong is by an infinite set of satisfying = In fact, the for this functor is to the set of all infinite over with the initial algebra solution in a complete category of infinite would the solution – in this case, there a solution in the because keeps if we work in a category of or consider partial one can obtain an initial solution representing but This that some functor yield infinite structures as fixed and the transfinite construction would a fixed point only in the limit the $\omega$-chain yields a of length which is The theory of with fixed points (like focus on initial fixed points yields defined structures (like In either case, the idea of a structure a fixed point of a functor is and Fixed In logic and the meaning of a set of recursive (a for is by the least fixed point of an operator on of (the immediate denoted for For a new from a set of assumptions starting from the set and iterating one approaches a limit where applying yields no new This limit is a fixed = By construction it is the least of the and complete the In categorical of = as an operator on the lattice of is and the theorem guarantees a least fixed point. 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It is also a case where transfinite steps are not needed $\omega$ or even steps if the theory is but one could consider transfinite for infinitely recursive in work by and Alpay a in language where a (the can an of semantic a process $\phi$ that represents $\phi$ an and or (like to semantic By applying $\phi$ to a piece of step it that the meaning converges to a stable of the In other words, if is a and then = it and as \to = approaches a fixed point where applying $\phi$ changes This is a a fixed point of the The $\phi^{\infty}$ (using the notation for the operator applied transfinitely many to denote the endpoint of this process. that a can recursive semantic and under infinite recursive the generative process converges to a fixed-point of $\phi^{\infty}$ here represents a semantic invariant of the – the meaning that remains after all self-referential is The existence of this fixed point and as a to by of This is a practical of a recursive fixed point in the transformation $\phi$ (a symbolic on or can be repeatedly applied to eventually yield a that $\phi$ Notably, the convergence to a fixed point provides guarantees of semantic in AI via Fixed In Alpay Algebra and Alpay a scenario where an AI and a until as a transfinite fixed-point The an state of a and the as an The transformation $\phi$ the AI its understanding on the and the (or in in a prove that by iterating this a functor on the state the system converges to a unique fixed a state where the of the is stable and the This fixed point is an essentially the understanding of the that change further Formally, if $\phi$ encapsulates one of for the is the limit of infinite a that satisfies = The show this convergence is and unique under category-theoretic conditions. This is a application of recursive fixed it provides a rigorous for AI by it as a fixed point in the semantic state of the The fixed point here an invariant meaning that the AI and the In terms, this at AI systems identity is a fixed point of – connecting to from theoretical computer science and that an identity could be defined as a stable fixed point of its self-referential from classical mathematics and AI demonstrate the of recursive fixed points. it is the of the natural number the semantics of a or the stable of an the is the a process that itself eventually reaches a point of that point, we have a fixed point that encapsulates the is to the of the fixed-point In the fixed point reached in $\omega$ steps iteration). In one a transfinite sequence of if the process is but the theory guarantees existence by transfinite The transfinite approach is a powerful – it even if a process as as each stage is built in a there is a fixed point at some ordinal work has even theory with transfinite fixed points Algebra to Banach’s contraction to transfinite In that a of an AI is to have an that is essentially a fixed point of a found via transfinite By Banach’s theorem to transfinite ordinal convergence to a unique semantic which is a recursive fixed point in a have a formal of recursive fixed emphasizing a rigorous By category theory and transfinite we an operator $\phi$ can a unique fixed point $\phi^{\infty}$ reached by an infinite recursive process. This fixed point exists under broad conditions or of and as a universal invariant for the process by $\phi$. We that $\phi^{\infty}$ is the smallest solution to $X = \phi(X)$ and that every other solution through its universal Our examples from mathematics numbers as fixed point, as fixed logic fixed point semantics of recursive and (iterative and semantic to stable In each case, the idea of a self-consistent fixed point of a recursive transformation provides and a on the of recursive fixed points and continuous processes. It a a system can be in of or self-referential one for the fixed point that represents the identity or Alpay’s recent further to that even an identity or a state can be as such a fixed point of its This a within formal mathematics – it that by the chain of transformations and one ensures the existence of a fixed point that encapsulates In practical terms, if every step of or transformation preserves or meaning without then as one iterates one approaches a state that cannot be further That state is the recursive fixed the point that and remains by its defining recursive fixed points are not just are the backbone of in self-referential By them, we that symbolic chains – as as are in a – will into a of This ensures that of symbols can be every is for in the limit, an fixed The mathematical here this in but the is through an recursive process, meaning and structure converge to an invariant fixed point, a on which further can Alpay Universal Alpay Algebra as Fixed-Point in and the of the in Alpay Algebra and the Fixed-Point of Alpay Algebra and Fixed-Point Lane, for the theorem and its of
This chapter is the first chapter of the section focused on the role of blockchain tokens in startup finance. The chapter is designed to lay the groundwork for understanding tokens and tokenomics. The chapter starts with basic token concepts such as defining tokens and classification of tokens (e.g. security vs. utility tokens, fungible vs. non-fungible tokens). Then, we discuss the roles tokens play on Web3 platforms. We conclude the chapter with a discussion of tokenomics and a case study.
A powerful feature in mechanism design is the ability to irrevocably commit to the rules of a mechanism. Commitment is achieved by public declaration, which enables players to verify incentive properties in advance and the outcome in retrospect. However, public declaration can reveal superfluous information that the mechanism designer might prefer not to disclose, such as her target function or private costs. Avoiding this may be possible via a trusted mediator; however, the availability of a trustworthy mediator, especially if mechanism secrecy must be maintained for years, might be unrealistic. We propose a new approach to commitment, and show how to commit to, and run, any given mechanism without disclosing it, while enabling the verification of incentive properties and the outcome—all without the need for any mediators. Our framework is based on zero-knowledge proofs—a cornerstone of modern cryptographic theory. Applications include both private-type settings such as auctions and private-action settings such as contracts, as well as non-mediated bargaining with hidden yet binding offers.
The widespread adoption of Artificial Intelligence (AI) has led to transformative advancements across industries such as healthcare, finance, supply chain, and smart governance.However, conventional AI systems are largely centralized, relying on siloed datasets and proprietary models controlled by a few entities.This centralized structure creates significant vulnerabilities, including data breaches, lack of transparency in decision-making, limited user control, and potential biases embedded within opaque algorithms.To address these limitations, this research investigates the integration of blockchain technology as a foundation for building decentralized intelligence.Blockchain, with its core properties of immutability, decentralization, and transparency, offers a compelling alternative to traditional AI deployment models.In this paper, we explore how blockchain can empower AI by decentralizing model training and data access, enabling tamper-proof audit trails, and fostering collaborative intelligence through smart contracts and distributed consensus mechanisms.Specific use cases such as decentralized federated learning, tokenized data marketplaces, and blockchain-governed AI agents are analyzed to illustrate practical implementations.We also examine the technical and ethical considerations of this convergence, including issues of scalability, interoperability, computational overhead, and regulatory compliance.Through a comprehensive review and conceptual framework, this paper contributes to the growing discourse on trustworthy and democratized AI, positioning blockchain as a key enabler of the next generation of secure, ethical, and transparent intelligent systems.
The main motivation of this thesis is the uncertain panorama of cybersecurity risks and threats, accentuated by the arrival of the quantum computer. This type of computer is completely disruptive, since its operation is governed by quantum mechanical phenomena. The implementation of Shors algorithm in a quantum computer with relevant size and performance will allow breaking the security of the most currently used pre-quantum asymmetric algorithms. This panorama makes it necessary to research new cryptographic paradigms that are resistant to quantum threats. Thus, quantum and post-quantum cryptography emerge. Several national security agencies are recommending the immediate migration to quantum-resistant solutions of vulnerable critical cryptosystems, mainly by implementing post-quantum algorithms, some of them recently standardized. Quantum cryptography bases its security on the same physical foundations as quantum computers, being independent of the computational capacity of an adversary. The implementation of solutions based on quantum cryptography still requires greater technological maturity, development of standards and certification of devices. In addition, the infrastructures necessary for these networks are expensive and difficult to scale, in their current conception, due to the need to have trusted intermediate nodes. However, the rapid advances in this field allow to further research quantum communications networks to be a reality for daily operations where a high level of security is required. The main objective of this thesis is to investigate quantum cryptography-based solutions that go beyond quantum key distribution (QKD). The thesis has focused on proposing two novel cryptographic mechanisms ensuring that the new protocols are comparable in efficiency with pre-quantum and post-quantum algorithms. Furthermore, it has been taken into account that these protocols are implementable in current quantum communications infrastructures (QCI) to maximize the technical benefit of the investments carried out for these deployments. As a result, a quantum-assisted digital signature protocol (Q-DS) and a quantum zero-knowledge proof (QZKP) have been proposed, analyzed and implemented, which combine symmetric pre-quantum mechanisms with QKD. The proposed quantum-assisted digital signature protocol avoids the use of vulnerable pre- quantum public-key cryptosystems, using symmetric keys generated by QKD and using them with widely known NIST-approved hash functions, giving rise to a composite cryptosystem whose security against various attacks is demonstrated. For its part, the proposed quantum zero-knowledge proof allows the authentication of users in a QCI without revealing personal information during the process. The proposal of a quantum version of ZKP has been done in this thesis for the very first time, without precedent in the literature. A theoretical study as well as experimental tests have been carried out, resulting in a secure and efficient authentication mechanism. Finally, given the industrial nature of this thesis, the evolution of the political panorama regarding quantum technologies and PQC have been closely followed, including the positions of relevant security-oriented organizations and economic investments for project funding. These issues, although not technical, have influenced the design of the cryptographic protocols proposed in this thesis. RESUMEN La principal motivación de esta tesis es el panorama incierto de los riesgos y amenazas de ciberseguridad, acentuado por la llegada del ordenador cuántico. Este tipo de ordenadores son completamente disruptivos, ya que su funcionamiento se rige por fenómenos mecánico-cuánticos. La implementación del algoritmo de Shor en un ordenador cuántico con tamaño y rendimiento relevantes permitirá romper la seguridad de los algoritmos asimétricos pre-cuánticos más utilizados actualmente. Este panorama hace necesario investigar nuevos paradigmas criptográficos que sean resistentes a las amenazas cuánticas. Así, surgen la criptografía cuántica y post-cuántica. Varias agencias de seguridad nacional han recomendado la migración inmediata de los criptosistemas críticos vulnerables a soluciones "quantum-resistant", principalmente mediante la implementación de algoritmos post-cuánticos, algunos de ellos recientemente estandarizados. La criptografía cuántica basa su seguridad en los mismos fundamentos físicos que los ordenadores cuánticos, siendo independiente de la capacidad computacional de un adversario. La implementación de soluciones basadas en criptografía cuántica aún requiere de mayor madurez tecnológica, desarrollo de estándares y certificación de dispositivos. Además, las infraestructuras necesarias para estas redes son costosas y difíciles de escalar, en su concepción actual, debido a la necesidad de contar con nodos intermedios de confianza. Sin embargo, los rápidos avances en este campo permiten que la investigación de las redes de comunicaciones cuánticas se vaya convirtiendo en una realidad para las operaciones diarias donde se requiere un alto nivel de seguridad. El objetivo principal de esta tesis es investigar soluciones basadas en criptografía cuántica que vayan más allá de la distribución de claves cuánticas (QKD). La tesis se ha centrado en proponer dos mecanismos criptográficos novedosos asegurando que los nuevos protocolos sean comparables en eficiencia con algoritmos pre-cuánticos y post-cuánticos. Además, se ha tenido en cuenta que estos protocolos sean implementables en las actuales infraestructuras de comunicaciones cuánticas (QCI) para maximizar el beneficio técnico de las inversiones realizadas para estos despliegues. Como resultado, se han propuesto, analizado e implementado un protocolo de firma digital asistido por claves cuánticas (Q-DS) y una prueba de conocimiento cero cuántica (QZKP), que combinan mecanismos pre-cuánticos simétricos con QKD. El protocolo de firma digital cuántica propuesto evita el uso de criptosistemas de clave pública pre-cuánticos vulnerables, utilizando claves simétricas generadas por QKD y utilizándolas con funciones hash ampliamente conocidas aprobadas por el NIST, dando lugar a un criptosistema compuesto cuya seguridad frente a diversos ataques se demuestra. Por su parte, la QZKP propuesta permite la autenticación de usuarios en una QCI sin revelar información personal durante el proceso. La propuesta de una versión cuántica de ZKP se ha realizado en esta tesis por primera vez, sin precedentes en la literatura. Se ha realizado un estudio teórico así como pruebas experimentales, dando como resultado un mecanismo de autenticación seguro y eficiente. Finalmente, dada la naturaleza industrial de esta tesis, se ha seguido de cerca la evolución del panorama político en relación con las tecnologías cuánticas y PQC, incluyendo las posiciones de las organizaciones relevantes en materia de seguridad y las inversiones económicas para la financiación de proyectos. Estas cuestiones, aunque no técnicas, han influido en el diseño de los protocolos criptográficos propuestos en esta tesis.
John C. Kolesar, Shan Ali, Timos Antonopoulos, Ružica Piskač
Zero-knowledge (ZK) protocols enable software developers to provide proofs of their programs’ correctness to other parties without revealing the programs themselves. Regular expressions are pervasive in real-world software, and zero-knowledge protocols have been developed in the past for the problem of checking whether an individual string appears in the language of a regular expression, but no existing protocol addresses the more complex PSPACE-complete problem of proving that two regular expressions are equivalent. We introduce Crêpe , the first ZK protocol for encoding regular expression equivalence proofs and also the first ZK protocol to target a PSPACE-complete problem. Crêpe uses a custom calculus of proof rules based on regular expression derivatives and coinduction, and we introduce a sound and complete algorithm for generating proofs in our format. We test Crêpe on a suite of hundreds of regular expression equivalence proofs. Crêpe can validate large proofs in only a few seconds each.
The rapid adoption of artificial intelligence (AI) systems, such as predictive AI, generative AI, and explainable AI, is in contrast to the slower development and uptake of robotic AI systems. Dynamic environments, sensory processing, mechanical movements, power management, and safety are inherent complexities of robotic intelligence capabilities that can be addressed using novel AI approaches. The current AI landscape is dominated by machine learning techniques, specifically deep learning algorithms, that have been effective in addressing some of these challenges. However, these algorithms are subject to computationally complex processing and operational needs such as high data dependency. In this paper, we propose a computation-efficient and data-efficient framework for robotic motion intelligence (RMI) based on vector symbolic architectures (VSAs) and blockchain-based smart contracts. The capabilities of VSAs are leveraged for computationally efficient learning and noise suppression during perception, motion, movement, and decision-making tasks. As a distributed ledger technology, smart contracts address data dependency through a decentralized, distributed, and secure transactions ledger that satisfies contractual conditions. An empirical evaluation of the framework confirms its value and contribution towards addressing the practical challenges of robotic motion intelligence by significantly reducing the learnable parameters by 10 times while preserving sufficient accuracy compared to existing deep learning solutions.
Anne Broadbent, Alex B. Grilo, Nagisa Hara, Arthur Mehta
In a proof of knowledge (PoK), a verifier becomes convinced that a prover possesses privileged information. In combination with zero-knowledge proof systems, PoKs play an important role in security protocols such as in digital signatures and authentication schemes, as they enable a prover to demonstrate possession of certain information (such as a private key or a credential), without revealing it. A PoK is formally defined via the existence of an extractor, which is capable of reconstructing the key information that makes a verifier accept, given oracle access to any accepting prover. We extend this concept to the setting of a single classical verifier and multiple quantum provers and present the first statistical zero-knowledge (ZK) PoK proof system for problems in QMA. To achieve this, we establish the PoK property for the ZK protocol of Broadbent, Mehta, and Zhao (TQC 2024), which applies to the local Hamiltonian problem. More specifically, we construct an extractor which, given oracle access to a provers' strategy that leads to high acceptance probability, is able to reconstruct the ground state of a local Hamiltonian. Our result can be seen as a new form of self-testing, where, in addition to certifying a pre-shared entangled state, the verifier also certifies that a prover has access to a quantum system, in particular, a ground state; this indicates a new level of verification for a proof of quantumness.
Yuming Huang, Jing Tang, Qianhao Cong, T. B. Richard · 6 authors
In blockchains using the Proof-of-Work (PoW) consensus mechanism, a mining pool is a joint group of miners who combine their computational resources and share the generated revenue. Similarly, when the Proof-of-Stake (PoS) consensus mechanism is adopted, the staking pool imitates the design of the mining pool by aggregating the stakes. However, in PoW blockchains, the pooling approach has been criticized to be vulnerable to the block withholding (BWH) attack. BWH attackers may steal the dividends from victims by pretending to work but making invalid contributions to the victim pools. It is well known that BWH attackers against PoW face the miner's dilemma . To our knowledge, despite the popularity of PoS, we are the first to study the pool BWH attack against PoS. Interestingly, we find that, for a network only consisting of one attacker pool and one victim pool, the attacker will eventually manipulate the network while the victim will vanish by losing the stake ratio gradually. Moreover, in a more realistic scenario with multiple BWH attacker pools and one solo staker who does not join any pools, we show that only one lucky attacker and the solo staker will survive, whereas all the other pools will vanish gradually, revealing the staker's dilemma . These findings indicate that, compared to PoW, the BWH attack on PoS has a much more severe impact due to the attacker's resource aggregation advantage. Our analysis is supported by experiments on massive real blockchain systems and numerical simulations.
Succinct arguments are proof systems that allow a powerful, but untrusted, prover to convince a weak verifier that an input x belongs to a language \(L \in \mathsf {NP}\) , with communication that is much shorter than the \(\mathsf {NP}\) witness. Such arguments, which grew out of the theory literature, are now drawing immense interest also in practice, where a key bottleneck that has arisen is the high computational cost of proving correctness. In this work, we address this problem by constructing succinct arguments for general computations, expressed as Boolean circuits (of bounded fan-in), with a strictly linear size prover. The soundness error of the protocol is an arbitrarily small constant. Prior to this work, succinct arguments were known with a quasi- linear size prover for general Boolean circuits or with linear-size only for arithmetic circuits, defined over large finite fields. In more detail, for every Boolean circuit \(C=C(x,w)\) , we construct an \(O(\log |C|)\) -round argument-system in which the prover can be implemented by a size \(O(|C|)\) Boolean circuit (given as input both the instance x and the witness w ), with arbitrarily small constant soundness error and using \(\mathrm{poly}(\lambda ,\log |C|)\) communication, where \(\lambda\) denotes the security parameter. The verifier can be implemented by a size \(O(|x|) + \mathrm{poly}(\lambda , \log |C|)\) circuit following a size \(O(|C|)\) private pre-processing step, or, alternatively, by using a purely public-coin protocol (with no pre-processing) with a size \(O(|C|)\) verifier. The protocol can be made zero-knowledge using standard techniques (and with similar parameters). The soundness of our protocol is computational and relies on the existence of collision resistant hash functions that can be computed by linear-size circuits, such as those proposed by Applebaum et al. (ITCS, 2017). At the heart of our construction is a new information-theoretic interactive oracle proof ( \(\mathsf {IOP}\) ), an interactive analog of a \(\mathsf {PCP}\) , for circuit satisfiability, with constant prover overhead. The improved efficiency of our \(\mathsf {IOP}\) is obtained by bypassing a barrier faced by prior \(\mathsf {IOP}\) constructions, which needed to (either explicitly or implicitly) encode the entire computation using a multiplication code.
We examine which decentralized finance architectures enable meaningful regulation by combining financial and computational theory. We show via deduction that a decentralized and permissionless Turing-complete system cannot provably comply with regulations concerning anti-money laundering, know-your-client obligations, some securities restrictions and forms of exchange control. Any system that claims to follow regulations must choose either a form of permission or a less-than-Turing-complete update facility. Compliant decentralized systems can be constructed only by compromising on the richness of permissible changes. Regulatory authorities must accept new tradeoffs that limit their enforcement powers if they want to approve permissionless platforms formally. Our analysis demonstrates that the fundamental constraints of computation theory have direct implications for financial regulation. By mapping regulatory requirements onto computational models, we characterize which types of automated compliance are achievable and which are provably impossible. This framework allows us to move beyond traditional debates about regulatory effectiveness to establish concrete boundaries for automated enforcement.
Secure multi-party computation is an area in cryptography which studies how multiple parties can compare their private information without revealing it. Besides digital protocols, many unconventional protocols for secure multi-party computation using physical objects have also been developed. The vast majority of them use playing cards as the main tools. In 2024, Kaneko et al. introduced the use of a balance scale and coins in zero-knowledge proof protocols for pencil puzzles. In this paper, we extend the use of these tools to secure multi-party computation. In particular, we develop four protocols that can securely compute any $n$-variable Boolean function using a balance scale and coins.
Abstraction Liquidity Theory (ALT) develops a formal framework for determining when local problem-solving traces become reusable abstraction assets that reduce downstream search, evaluation, and certification costs. The paper treats abstractions as operational tokens rather than informal artifacts, and evaluates them through declared receivers, opportunity measures, baselines, lifecycle costs, telemetry, evidence validity, transport scope, authority envelopes, hazard constraints, and runtime certificate packets. The manuscript introduces an actor-neutral certification kernel for AI agents and other computational actors. It specifies machine-readable packet schemas, dual exploration and settlement ledgers, finite-sample lower and upper bounds, causal and calibrated-proxy value estimands, mission-validity certificates, adversarial-token rejection, root/finality checks, baseline refresh, deprecation, resurrection, rollback, and kernel-update bridges. The goal is to make abstraction evaluation executable: an agent should be able to parse a packet, verify evidence, admit or reject a token, suspend stale claims, deprecate negative-liquidity tokens, and preserve raw net safe capital under fail-closed rules. The paper further defines Target-valid ALT-CARA, a criterion for certified ASI realization acceleration. Rather than claiming unconstrained ASI achievement, ALT-CARA formalizes time-to-target acceleration relative to a resource-matched baseline upper envelope, under declared capability bases, target-validity certificates, raw net solvency, viability conditions, hazard and authority constraints, transport validity, finality, and causal reproduction evidence. The framework connects AI evaluation, causal inference, runtime verification, risk control, skill reuse, safe exploration, and distributed certification into a single theory of mission-valid safe abstraction capital.
Orestis Melkonian, Wouter Swierstra, James Chapman, Sub Software Technology · 6 authors
Distributed ledgers nowadays manage substantial monetary funds in the form of cryptocurrencies such as Bitcoin, Ethereum, and Cardano. For such ledgers to be safe, operations that add new entries must be cryptographically sound - but it is less clear how to reason effectively about such ever-growing linear data structures. This paper demonstrates how distributed ledgers may be viewed as computer programs, that, when executed, transfer funds between various parties. As a result, familiar program logics, such as Hoare logic, are applied in a novel setting. Borrowing ideas from concurrent separation logic, this enables modular reasoning principles over arbitrary fragments of any ledger. All of our results have been mechanised in the Agda proof assistant.
Zero-Knowledge Proofs (ZKPs) are public key cryptosystem that enables to demonstrate that a statement which is known by them is correct without revealing the same to the verifier. ZKPs have moved in modern cryptographic systems, blockchain applications, decentralized finance (DeFi) and identity authentication systems. This paper explores the evolution of ZKPs and their significance as in secure and privacy preserving. We classify ZKPs into two groups namely interactive and non-interactive, discussing prominent protocols such as zk-SNARKs, zk-STARKs, Bulletproofs, PLONK, and Halo2. Each approach has advantages as efficiency, proof size, and computational overhead. The study further examines the multitude of applications of ZKPs, as privacy-enhanced blockchain transactions, zero-knowledge rollups for scalability, decentralized identity management, secure voting mechanisms, and regulatorycompliant financial systems. With advantages, possible limitations in scalability, lack of standardization, and vulnerabilities to emerging quantum computing threats. Due to the restrictions, hardware acceleration through GPUs and others, presents promising solutions, while new protocols such as PLONK and Halo2 seek to optimize performance to earlier developed solutions. Finally, we discuss the future trajectory of ZKPs. This review aims to provide an understanding of the current state of ZKP research, its applications, and the key challenges that need to be addressed to facilitate broader adoption.